I am trying to learn DPS. A couple of explanations are based on statistics. I would like to understand what is and how coherence works, but I am stuck on its definition.

I found the following definition of coherence: $$C(\omega)=\frac{S_{XY}(\omega)}{\sqrt{S_{XX}(\omega)S_{YY}(\omega)}}$$ while a definition of $S_{XY}(\omega)$ is related to cross spectrum by DFT by the formula: $$r_{xy}[k]=E\{x[n]y[n-k]\}$$ How to understand expected value here? Unfortunately the explanation I found, e.g. on Wikipedia, is based on probability: $$E[X]=\sum_{i=1}^{\inf}p_ix_i$$ I understand its simple examples such as flipping a coin or throwing or rolling dice etc. When I look at this definition: $E\{x[n]y[n-k]\}$, I don't know where the probability term is.

  • $\begingroup$ Can you fill in some details here: relate $r_{xy}$ to $S$ and provide the definitions of $x[n]$ and $y[n]$ $\endgroup$
    – user237392
    Commented Jun 22, 2015 at 12:24
  • $\begingroup$ Apparently $X$ and $Y$ are random variables, so the expectation is with respect to their joint distribution. $\endgroup$
    – dsaxton
    Commented Jun 22, 2015 at 15:13
  • $\begingroup$ $x$ and $y$ are simply time domain signals in signal processing. $r_{xy}$ and $S_{XY}$ are related by Fourier Transform (DFT particularly). $\endgroup$
    – Celdor
    Commented Jun 23, 2015 at 7:07


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